By Ernst Levy, Siegmund Levarie
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Additional info for A theory of harmony
For our purpose the index 16 is sufficient. : The tones 1/7, 1/11, 1/13 are not used in our tone system. Therefore we have no name for them and can describe them only by referring them to the nearest known tone. The signs ÙandÚ indicate that they are, respectively, higher or lower than the note referred to. We write the series on graph paper, in one line from left to right, using one square (inch or centimeter) for each tone, and next proceed to multiply the string by the same series of digits.
Thus it is but natural that we should choose the series of digits for our guiding principle. Again, after we have made that decision, two roads of approach seem to be available. We could divide and multiply the string according to the series of whole numbers and note the resulting tone relations, or we could start from the tone relations we know (intervals) and observe the corresponding string relations. The end result would be the same, but the first method recommends itself for its greater consistency and objectivity.
A set of moveable bridges will divide the strings in any desired proportion. These bridges (triangular wooden blocks will do) should exceed the distance between the board and the string by only a very small amount, in order to avoid sizeable additional tension which would change the pitch of the tone. Graph paper tacked on the board makes measurements easy and permits convenient recording of results. In sonometer experiments, absolute pitch is irrelevant, as we are interested only in tone relations.